Prove that the greatest integer function `f: R rarr R`, given by `f(x)=[x]`, is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Explanation

Given `f(x)=[x]`


It is seen that `f(1.2)=[1.2]=1,f(1.9)=[1.9]=1`


`therefore``f(1.2)=f(1.9)`, but `1.2!=1.9`

`therefore`f is not one-one

Now, consider `0.7inR`

Now, since `f(x)=[x]` is always an integer.
Thus, there does not exist any element `x∈R ` such that `f(x)=0.7`

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